betta in a 3 gal

Yes and no. I use gravel and filter media from another tank so I don't have to cycle my tank. I would not put a betta into an un-cycled tank unless I was prepared to do 100 % water changes.

I always cycle every one of my tanks, there is no need to put the fish through suffering if you can prevent it.

Bettas are tropical fish too. Cycle, heat, filter, non snagging decorations...for me that means live plants and a piece of driftwood.

u cycle for every tank you set-up. betta's included. if you dont want to do this process keep a filter running extra on another tank so you dont have to cycle.

six weeks id do but mabye shorter

I would just let it run for 2-3 weeks while you dose ammonia on the tank you want cycled.

Is your 55 gallon cycled yet? If so, you can take some gravel or some filter media and add it to the tank as you add the betta. Or, you can do as others suggested and run the new filter on the cycled tank for a week or two, then move it to the betta tank at the same time that you add the betta. You will still need to test the water daily in the betta tank until you are sure it has cycled... this isn't always an instant cycle, but will certainly be much abbreviated. Be prepared to change some water.

Or, if you want to do a fishless cycle, check out the article here on cycling. You can dose ammonia (with no surfactants) to cycle your tank with no fish. It's really easy, and the process works the same way as a fishy cycle (ammonia spike, then nitrite spike, then nitrates), but is usually faster. You can also skip the ammonia and just chuck a raw shrimp from the grocery store in a filter bag (or a nylon stocking) and let it rot. It's nasty, but extremely easy, lol.